Covariant constraints in ghost free massive gravity

TL;DR

The paper develops a vielbein formulation of ghost-free massive gravity (dRGT) to count covariant constraints and explain the absence of the Boulware-Deser ghost. It shows that, starting from 16 dynamical vierbein components, local Lorentz and diffeomorphism invariances remove $10$ constraints, leaving $6$ degrees of freedom, with an extra scalar constraint arising only for specific combinations of mass terms $\beta_n$ (e.g., $\beta_1\neq0$ or $\beta_2\neq0$). The authors derive this extra constraint by exploiting Bianchi identities and traces of the field equations, and they discuss cases where a universal scalar constraint is not straightforward to obtain (notably when $\beta_3\neq0$). They also present a Stueckelberg-free method to read off kinetic terms for the massive graviton’s extra polarization via a simple shift in the vierbein. Together, these results illuminate why certain dRGT models are ghost-free and provide a covariant, Lagrangian route to the extra constraint, consistent with Hamiltonian analyses.

Abstract

We show that the reformulation of the de Rham-Gabadadze-Tolley massive gravity theory using vielbeins leads to a very simple and covariant way to count constraints, and hence degrees of freedom. Our method singles out a subset of theories, in the de Rham-Gabadadze-Tolley family, where an extra constraint, needed to eliminate the Boulware Deser ghost, is easily seen to appear. As a side result, we also introduce a new method, different from the Stuckelberg trick, to extract kinetic terms for the polarizations propagating in addition to those of the massless graviton.